Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.463 - 0.885i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.750 − 0.661i)3-s + (0.309 − 0.951i)7-s + (0.125 − 0.992i)9-s + (0.338 − 0.940i)11-s + (0.612 + 0.790i)13-s + (0.770 − 0.637i)17-s + (0.750 + 0.661i)19-s + (−0.397 − 0.917i)21-s + (0.876 + 0.481i)23-s + (−0.562 − 0.827i)27-s + (−0.975 − 0.218i)29-s + (−0.637 − 0.770i)31-s + (−0.368 − 0.929i)33-s + (0.827 + 0.562i)37-s + (0.982 + 0.187i)39-s + ⋯
L(s,χ)  = 1  + (0.750 − 0.661i)3-s + (0.309 − 0.951i)7-s + (0.125 − 0.992i)9-s + (0.338 − 0.940i)11-s + (0.612 + 0.790i)13-s + (0.770 − 0.637i)17-s + (0.750 + 0.661i)19-s + (−0.397 − 0.917i)21-s + (0.876 + 0.481i)23-s + (−0.562 − 0.827i)27-s + (−0.975 − 0.218i)29-s + (−0.637 − 0.770i)31-s + (−0.368 − 0.929i)33-s + (0.827 + 0.562i)37-s + (0.982 + 0.187i)39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.463 - 0.885i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.463 - 0.885i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.463 - 0.885i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (1333, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ -0.463 - 0.885i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.210103673 - 3.651471989i$
$L(\frac12,\chi)$  $\approx$  $2.210103673 - 3.651471989i$
$L(\chi,1)$  $\approx$  1.488943043 - 0.7540309519i
$L(1,\chi)$  $\approx$  1.488943043 - 0.7540309519i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.39755004515244447111156618197, −18.08582827474757215752094884811, −17.087539718931504174785564954893, −16.39730160795060149176715151414, −15.62839800438188037671676588088, −14.99926606273887302589703964405, −14.76379952304062444515973528230, −13.9190267702404133371293344990, −12.940353544006843591115818578918, −12.58827910914832910141769130645, −11.56582210228335289795026728859, −10.88058017594155464663344158262, −10.18445298911261826326548702358, −9.32562775579097507145281242339, −8.989127621486148730738084273409, −8.126829080869073007795600016878, −7.565822173538775441275981702931, −6.64015713714866966414853559957, −5.44753795927784361032187698522, −5.21065148967970505511893677661, −4.17537332297283888676425054889, −3.430187507678454920219661068897, −2.70425831567087894222178133, −1.93036983685961597096964435469, −1.019323255008147626229550750878, 0.625544890776639717893098145, 1.14499445578610224161181729525, 1.92674361532172023880720070243, 3.06720272987894603892825789250, 3.64295498476279722362901494235, 4.251302015650215852031366030067, 5.50863417671463296445703629070, 6.13634214577170398895178260162, 7.16449433729088753497974343545, 7.46926955342537372011444861371, 8.24096241248848573581268139144, 9.077762693675396348324909206278, 9.539494170125627053099443114445, 10.50735349425248139038835699436, 11.48337422642260456556966593654, 11.68668036552201201460968066122, 12.85555562232995142372357926429, 13.46040919440165767801321879529, 13.98867983110116125797389498045, 14.41591082936532250978416327862, 15.19974465411092517130185637577, 16.2641930044990632780531666723, 16.6559041776644185122726324453, 17.43071203406968042222399637942, 18.311334877173953265236670868861

Graph of the $Z$-function along the critical line